Question: Simplify and expand the following expression: $ \dfrac{5q + 3}{q + 3}+\dfrac{5q - 1}{4q + 3} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(q + 3)(4q + 3)$ Multiply the first term by $\dfrac{4q + 3}{4q + 3}$ $ \begin{align*} \dfrac{5q + 3}{q + 3} \times \dfrac{4q + 3}{4q + 3} & = \dfrac{(5q + 3)(4q + 3)}{(q + 3)(4q + 3)} \\ & = \dfrac{20q^2 + 27q + 9}{(q + 3)(4q + 3)}\end{align*} $ Multiply the second term by $\dfrac{q + 3}{q + 3}$ $ \begin{align*} \dfrac{5q - 1}{4q + 3} \times \dfrac{q + 3}{q + 3} & = \dfrac{(5q - 1)(q + 3)}{(4q + 3)(q + 3)} \\ & = \dfrac{5q^2 + 14q - 3}{(4q + 3)(q + 3)}\end{align*} $ Now we have: $ = \dfrac{20q^2 + 27q + 9}{(q + 3)(4q + 3)} + \dfrac{5q^2 + 14q - 3}{(4q + 3)(q + 3)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{20q^2 + 27q + 9 + 5q^2 + 14q - 3}{(q + 3)(4q + 3)} $ $ = \dfrac{25q^2 + 41q + 6}{(q + 3)(4q + 3)}$ Expand the denominator: $ = \dfrac{25q^2 + 41q + 6}{4q^2 + 15q + 9}$